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Variational Inference Using Material Point Method

Yongchao Huang

TL;DR

This work introduces MPM-ParVI, a gradient-based, physics-guided sampling method that casts variational inference as the deformation of an interacting particle system via the Material Point Method. By representing the target density with a score field and applying it as an external force on a background grid, the method deterministically evolves particles to approximate $p(\mathbf{x})$ through a structured P2G–G2P cycle with PIC or APIC transfers. The approach integrates continuum mechanics with probabilistic inference, enabling automatic mass/momentum conservation, multi-modality handling, and efficient simulation of large deformations, while remaining easy to implement and parallelizable. However, it faces scalability challenges in high dimensions due to the intrinsic grid-based representation and requires further theoretical grounding and empirical validation. Overall, MPM-ParVI broadens the spectrum of physics-based ParVI methods, offering a novel, deterministic pathway for Bayesian inference and score-based generative modelling that leverages the strengths of MPM for complex densities and geometric configurations.

Abstract

A new gradient-based particle sampling method, MPM-ParVI, based on material point method (MPM), is proposed for variational inference. MPM-ParVI simulates the deformation of a deformable body (e.g. a solid or fluid) under external effects driven by the target density; transient or steady configuration of the deformable body approximates the target density. The continuum material is modelled as an interacting particle system (IPS) using MPM, each particle carries full physical properties, interacts and evolves following conservation dynamics. This easy-to-implement ParVI method offers deterministic sampling and inference for a class of probabilistic models such as those encountered in Bayesian inference (e.g. intractable densities) and generative modelling (e.g. score-based).

Variational Inference Using Material Point Method

TL;DR

This work introduces MPM-ParVI, a gradient-based, physics-guided sampling method that casts variational inference as the deformation of an interacting particle system via the Material Point Method. By representing the target density with a score field and applying it as an external force on a background grid, the method deterministically evolves particles to approximate through a structured P2G–G2P cycle with PIC or APIC transfers. The approach integrates continuum mechanics with probabilistic inference, enabling automatic mass/momentum conservation, multi-modality handling, and efficient simulation of large deformations, while remaining easy to implement and parallelizable. However, it faces scalability challenges in high dimensions due to the intrinsic grid-based representation and requires further theoretical grounding and empirical validation. Overall, MPM-ParVI broadens the spectrum of physics-based ParVI methods, offering a novel, deterministic pathway for Bayesian inference and score-based generative modelling that leverages the strengths of MPM for complex densities and geometric configurations.

Abstract

A new gradient-based particle sampling method, MPM-ParVI, based on material point method (MPM), is proposed for variational inference. MPM-ParVI simulates the deformation of a deformable body (e.g. a solid or fluid) under external effects driven by the target density; transient or steady configuration of the deformable body approximates the target density. The continuum material is modelled as an interacting particle system (IPS) using MPM, each particle carries full physical properties, interacts and evolves following conservation dynamics. This easy-to-implement ParVI method offers deterministic sampling and inference for a class of probabilistic models such as those encountered in Bayesian inference (e.g. intractable densities) and generative modelling (e.g. score-based).
Paper Structure (32 sections, 65 equations, 2 figures, 3 tables, 3 algorithms)

This paper contains 32 sections, 65 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Meshless particle simulation methods
  4. ParVI methods
  5. MPM-based sampling: methodology
  6. MPM formulation
  7. The interpolation function
  8. Representing the target score as external body force
  9. Evolving the IPS
  10. Discussions
  11. Lagrangian and Eulerian descriptions of motion
  12. Time step size
  13. Multi-modal inference
  14. Memory and complexity
  15. Advantages of MPM-ParVI
  16. ...and 17 more sections

Figures (2)

  • Figure 1: MPM description of motion of a continuum body in 3D space (figure from Kumar2022). $\Omega^0$ and $\Omega^t$ are the initial and current configurations of the whole material domain, respectively. $p$ represents a material point (i.e. a particle). $\mathbf{X}^0$ and $\mathbf{x}^t$ are the initial and current positions of the material point, respectively. $\phi_t$ is the mapping function between initial and current configurations.
  • Figure 2: One MPM-ParVI cycle illustrating Algo.\ref{['algo:MPM_sampling1']} and Algo.\ref{['algo:MPM_sampling2']}. Red circles represent material points overlaid on a regular mesh grid with nodes represented by small squares. P2G: arrows represent particle property states (mass and momentum) being projected (aggregated) onto grid nodes. Grid updating: equations of motion are solved on grid nodes, resulting in updated nodal velocities. G2P: arrows represent the updated nodal property states (nodal velocities) being mapped back to particles. Finally, grid is reset. Figures are modified from DeVaucorbeil2020, equations within dashed boxes are only used in Algo.\ref{['algo:MPM_sampling2']}.